MHB How to derive $P(PH)$ without using a joint distribution table?

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To derive $P(PH)$ without a joint distribution table, the sum rule and product rule can be applied. Using the given probabilities, $P(PH)$ can be calculated as $P(PH | H)P(H) + P(PH | \lnot H)P(\lnot H)$. Substituting the values, $P(PH | H) = 0.8$, $P(PH | \lnot H) = 0.3$, $P(H) = 0.1$, and $P(\lnot H) = 0.9$, results in $P(PH) = 0.8 \times 0.1 + 0.3 \times 0.9$. This calculation yields a final result for $P(PH)$.
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Given $P(PH | H) = 0.8$ and $P(PH | \lnot H) = 0.3 $ and $P(H) = 0.1$ how can I derive $P(PH)$ without resorting to a joint distribution table?
 
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tmt said:
Given $P(PH | H) = 0.8$ and $P(PH | \lnot H) = 0.3 $ and $P(H) = 0.1$ how can I derive $P(PH)$ without resorting to a joint distribution table?

Hi tmt,

We can use that (applying sum rule and general product rule):
$$P(A) = P((A\land B)\lor (A\land \lnot B)) = P(A\land B) + P(A\land\lnot B) = P(A\mid B)P(B) + P(A\mid \lnot B)P(\lnot B)$$
 
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